Substantia. An International Journal of the History of Chemistry 7(1): 15-22, 2023 Firenze University Press www.fupress.com/substantia ISSN 2532-3997 (online) | DOI: 10.36253/Substantia-1872 Citation: Krishtalik L.I. (2023) The Rate Constant – Reaction Free Energy Dependence for the Electron Trans- fer Reactions in Solutions. The Way to Interpret the Experimental Data Correctly. Substantia 7(1): 15-22. doi: 10.36253/Substantia-1872 Received: Jul 03, 2022 Revised: Feb 05, 2022 Just Accepted Online: Feb 08, 2022 Published: March 13, 2023 Copyright: © 2023 Krishtalik L.I. This is an open access, peer-reviewed article published by Firenze University Press ( h t t p: // w w w.f u p r e s s .c o m /s u b s t a n - tia) and distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All rel- evant data are within the paper and its Supporting Information files. Competing Interests: The Author(s) declare(s) no conflict of interest. Research Article The Rate Constant – Reaction Free Energy Dependence for the Electron Transfer Reactions in Solutions. The Way to Interpret the Experimental Data Correctly Lev I. Krishtalik†1 A.N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Moscow For correspondence please contact Dr. Vasily Ptushenko (ptush@mail.ru) 1Lev I. Krishtalik left a noticeable mark in electrochemistry and biophys- ics. His scientific career started in the mid-1940s, when the hardships of war- time were replaced first by hopes for the best, and then by persecution of the numerous segments of the population and many scientific schools in the USSR. † 23.11.1927–26.02.2022 Lev I. Krishtalik. Courtesy of Vasily Ptushenko. http://www.fupress.com/substantia http://www.fupress.com/substantia http://www.fupress.com/substantia 16 Lev I. Krishtalik Fortunately, Lev Krishtalik survived these difficult years, although he had to work for 15 years in the industry, doing science only in his off-duty hours. Even in these hard conditions, he managed to perform an outstanding work on the experimental detection of barrierless elec- trode reactions (discharge of hydrogen ions). Later, he worked in the Alexander N. Frumkin Institute and con- tributed substantially to the theory of charge transfer reactions developed by Revaz R. Dogonadze and Alex- ander M. Kuznetsov and to its experimental confirma- tion. This was a cutting-edge field in chemistry and led to Rudolph A. Marcus’ Nobel Prize in Chemistry in 1992. A little later, Lev Krishtalik developed his views on the cata- lytical abilities of enzymes. The work of Krishtalik was well known at the time and his efforts were rewarded with the Frumkin Prize of the International Society of Electrochemistry in 2001. Note that he was a disciple and one of the closest col- leagues of Alexander N. Frumkin, although he was never formally his student. Along with his scientific interests, Lev Krishtalik also had interests and encyclopedic knowledge in different cul- tural fields, including poetry, music, painting, and archi- tecture. One of his favorite cities was Florence, and the painting “La Catena” with the medieval/renaissance land- scape of Florence was always on his bookshelf. His per- sonal memoirs have been recently published (in Russian; https://7i.7iskusstv.com/avtory/krishtalik/). Lev Krishtalik was a radiant personality. Even in the most difficult moments of his life, he remained friendly to people and grateful to fate. This is the last manuscript which he worked on in his two or three last years of his life. Unfortunately, his illness did not allow him to publish the paper before his death. Vasily Ptushenko Abstract. The relative influences of the reorganization energies of the classical and quantum modes on the maximum position of the rate constant – reaction free energy curve have been studied. In the framework of the continuum electrostatics, the electron transfer reorganization energies in methyltetrahydrofurane solutions for the system biphenylyl – spacer – acceptor were calculated. For different acceptors, the solvent reorganization energy varies from 1.0 to 1.1 eV. When added with the rather small reorganiza- tion energies for classical intra-molecular modes we obtain 1.13–1.34 eV. With account of possible errors this coincides practically with the experimental estimate of the energy at the maximum of the rate–free energy curve ΔGmax ≈ -1.2 eV. Hence, we can con- clude that the reorganization of quantum modes does not influence substantially the position of this maximum. To the contrary, in a non-polar solvent isooctane were the solvent reorganization does not play any role the reorganization of the quantum intra- reactants modes becomes determinant. These conclusions agree fully with the results of the general theoretical analysis and should be accounted for in the experimental data interpretation. Keywords: Reorganization energy, Medium reorganization, Intra-molecular reorganization, Rate maximum. List of Abbreviations: A – acceptor, BPH – biphe- nylyl, D – donor, MTHF – metyltetrahydrofurane, Py – pyrenyl, Sp – spacer, Q – benzoquinonyl. 1. INTRODUCTION One of the fundamental problems of the chemical and electrochemical kinetics is the physical mechanism of the elementary act of the electron transfer, especial- ly in condensed media. This question was intensively studied in many theoretical works (in particular [1-4]; a short review of them is given in Section 2. 1). Electron transfer is connected with some reorganization of the reactant’s polar surroundings and reorganization of the molecules inner structure. These two processes are char- acterized by the corresponding parameters – the medi- um (solvent) reorganization energy λs and the intramo- lecular reorganization energy λv. The relative contribution of these processes has been many times discussed. In liquid solutions the kinetics of the electron transfer proper can be distorted by kinetics of the mutual diffusion of the reactants. Therefore, most suitable for the comparison of the experimental results with the theory are the systems in which the donor and the acceptor are connected by a rigid linker ensuring a constant distance between the reactants. In the pre- sent paper, a typical example of such a system will be considered. In many papers dealing with the problem in chemical and biochemical systems [5-9], it has been a priori accepted that both the parameters – λs and λv – contribute substantially to the final result. Accordingly, both parameters were found by fitting the experimental data to the theoretical equation derived under the same assumption. The results of such a fitting are not quite unequivocal. Therefore, it is highly desirable to estimate the reorganization energies, both λs and λv by independ- ent methods. As a vivid illustration of the possible prob- 17The Rate Constant – Reaction Free Energy Dependence for the Electron Transfer Reactions in Solutions lem the data of [7] can be mentioned. In this paper, the fitting for the system porphyrines – C2H4 – quinones in benzene and toluene has been performed with two sets of parameters – the first one λs = 0.18 eV, λV = 0.60 eV and the second λs = 0.60 eV, λV = 0.20 eV. The second set which is physically quite unrealistic gives the results almost so good as the first one. This shows again that one should use parameters obtained from the independent data. To emphasize this conclusion is the aim of the pre- sent paper. Section 2 deals with the theory (2.1) and calcula- tions (2.2); in Section 3, the results described and dis- cussed; in Section 4, the conclusion are presented. 2. THEORY AND CALCULATIONS. 2.1 Theory In this Subsection, a short description of the most important theoretical results used in this paper is pre- sented. Marcus [1] was the first who has given, in the frame- work of a semi-classical theory, the correct exponential dependence of the reaction rate on the reaction free energy ΔG and the reorganization energy λ. Levich and Dogonadze [2] have done a quantum–mechanical analy- sis of the problem. They have obtained the same expo- nent and a strict expression for the pre–exponential fac- tor. The final result is (1) Here k is the rate constant, V(r) – the distance r dependent electronic coupling matrix element, h – Planck constant, kB – Boltzmann constant, T - tempera- ture. This equation is valid only at the high-temperature limit, e.g. under condition ћw << kBT, where ћw is the energy of vibration coupled to the electron transfer. From Eq.1 follows that the rate constant – reaction free energy dependence presents a symmetric parabola with maximum at -ΔGmax=λ (1a) For the purpose of qualitative treatment of results it is instructive to recall the Marcus formula for the medi- um reorganization energy (2). (2) Here e is the charge to be transferred, εo and εs are optical and static dielectric permittivities of the medi- um, a1 and a2 are the radii of the spherical reactants, and R is the inter-center distance. This simple analytical solution is obtained for the case of two spherical reac- tants in an infinite homogenous medium. At a more complex geometry and an inhomogeneous medium one employs the numerical methods of calculation (section 2.2) with the same physical rationale as for derivation of the Marcus Eq.2; hence the latter is suitable for the qual- itative discussion. Dogonadze et al. [3] have considered a more general case when different oscillators are present with a wide range of the frequencies. They have shown the principal difference in the behavior of the intramolecular vibra- tions that have a high frequency ћω>>kT, i.e. much more than own vibrations of liquid solvent (ћω< |ΔGmax|, e.g. in the inverted region, the excited vibration levels come into play mak- ing k – DG curve asymmetric. 3.2. BPH–Sp–A in isooctane. The data on this system are presented according to [14] at Fig. 1. The dashed curve is tacked from [14]. It is calculated with the following parameters: λs = 0.15 eV 20 Lev I. Krishtalik and λv= 0.45eV. Strictly speaking, in the nonpolar sol- vent isooctane λs = 0. However, with account of the tor- sional component the accepted value is reasonable for the classical reorganization energy. For the vibrational component the authors quite logical accepted the same value as was obtained previously by fitting the data for the same system in MTHF. However, as it is clear from Fig.1 the agreement of this calculation with the experi- ment can be hardly considered as satisfactory one. The deviation of the experimental points from the theoreti- cal curve in some cases can be as large as ~ 4 orders of magnitude while for MTHF there is only for one experi- mental point with the maximal deviation not exciding 0.3 orders. Moreover, the experimental data cannot be described as a bell-shaped dependence. Therefore, it is hardly possible to establish definitely the value of ΔGmax. The reason for that lies, most probably, just in the mechanism of the elementary act of the electron trans- fer. In absence or at quite small classical reorganization energy the initial and final electronic energy levels can be equalized only at the expense of the excitation of intra-molecular vibrations. Just this idea is accounted for in Jortner equation. However, this equation in fact does not imply the same λs and also the same total λv for all acceptors. However, as it is shown in Section 3.2 for the different reactants λs and, especially, λv can be substan- tially different. This is very likely the reason of the dis- crepancies between experiment and calculations repre- sented on Fig. 1. Two additional remarks seem to be proper. First, in the presence of several modes the total λv equals to the sum of the corresponding parameters for each mode (the model calculations of Ulstrup and Jortner [25]). Second, there exists a wide set of less effective modes, forming a quasi- continual spectrum important at gradual excitation. Note also that the difference in the intra-molecular quantum frequencies does not influence the data for MTHF because in polar liquid solvent the location of the rate maximum is determined by the classical reorganiza- tion only. 4. CONCLUSION Nowadays, experimental data processing often uses Jortner formula which implies a substantial contribution of both reorganization energies (of the intra-molecular quantum modes and the classical medium modes) into the total reorganization energy. The latter determines the reaction free energy corresponding to the maximum reaction rate. However, the values of the two reorgani- zation energies are usually taken not from some inde- pendent sources but obtained by fitting the kinetic data to Jortner equation. Correspondingly, there is not any attempt to prove the basic condition of a comparable effect of the two reorganization energies. The specific feature of the approach advanced in this paper is determination of all the parameters not by fitting the kinetic data but basing on the independent experimental data. The medium reorganization energy has been calculated electrostatically. For this purpose, not the simplified Marcus formula implying spherical shape of the reactants but the more general method of the numerical solution of the Poisson–Boltzmann equa- tion was employed. This method allows accounting for the real shape and size of the reactants. The calculations performed for solutions in a polar solvent MTHF show that the medium reorganization gives the predominant contribution to the total reorganization energy. To the contrary, in a non–polar isooctane, the medium reor- ganization does not play any substantial role, and the intra-molecular reorganization becomes predominant. In the latter case, the shape of the rate – free energy curve is rather complex due to differences in vibration spectra of various molecules. All these results are in agreement with the general theoretical conclusions. From the above, the algorithm for processing experimental data follows. In the case of reaction in polar solvent one should perform a strict electrostatic Figure 1. System biphenylyl–spacer–acceptor in isooctane solu- tions. Dashed line correspond to calculations [14] with the follow- ing parameters: λs = 0.15 eV, λv = 0.45 eV, ω = 1500 cm-1, V = 6.2 cm-1. Dotted line is just a guide for eye that is drown according to the sequence of the reaction free energies. The symbols indicate experimental data: 1— 4-biphenylyl, 2— 2-naphthyl, 3— 9-phen- anthryl, 4— 1-pyrenyl, 5— 2-(5,8,9,IO-tetahydronaphthoquinonyl), 6— 2-naphthoquinonyl, 7— 2-benzoquinonyl, 8— 2- (5-chloroben- zoquinonyI), 9— 2-(5,6-dichlorobenzoquinonyl). 21The Rate Constant – Reaction Free Energy Dependence for the Electron Transfer Reactions in Solutions calculation of the medium reorganization energy and try to analyze the other possible classical modes. For the reaction in a non-polar medium one should analyze the vibration spectra of the reactants and calculate the cor- responding reorganization energies. ACKNOWLEDGEMENT I am indebted to Prof. An.M. Kuznetsov for provid- ing me the quantum chemical data on the partial charg- es distribution in quinone, and to Dr. V.V. Ptushenko for his help in preparing the paper for publication. REFERENCES [1] R.A. Marcus (1956) On the theory of oxidation- reduction reactions involving electron transfer. I. J Chem Phys 24 966–978. [2] V.G. 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Jortner (1980) Dynamics of electron transfer in bacterial photosynthesis. Biochim Biophys Acta 594 193–230. [23] R. Borrelli, W. Domcke (2010) First-principles study of photoinduced electron-transfer dynamics in a Mg–porphyrin–quinone complex. Chemical Physics Letters, 498 230–234. [24]. S.F. Fischer, R.P. Van Duyne (1977) On the theory of electron transfer reactions. The naphtalene-./ TCNQ system. Chemical Physics 26 9–16. [25] J. Ulstrup, J. Jortner (1975) The effect of intramo- lecular quantum modes on free energy relationships for electron transfer reactions. J. Chemical Physics 63 4358–4368. Substantia An International Journal of the History of Chemistry Vol. 7, n. 1 – 2023 Firenze University Press Superbugged Pierandrea Lo Nostro Equivalence of Electromagnetic Fluctuation and Nuclear (Yukawa) Forces: the π0 Meson, its Mass and Lifetime Barry W. 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